outputs, a direct optical comparison of the
clocks was made. This was performed by com-
bining the optical clock signals onto a single
photodetector, directly generating an electrical
signal at an intentionally offset beat frequency
between the clocks. This beat frequency was
also digitally sampled and recorded. Compar-
ing the phase of the difference frequency of the
microwave outputs to that of the optical beat
frequency allowed for confirmation of high-
fidelity phase and frequency transfer to the
electronic domain.
It is interesting to note the level of resolu-
tion required to measure a fractional frequency
instability of 10−^16 at 1 s on a 10-GHz signal.
This implies tracking phase changes cor-
responding to only one-millionth of a cycle. As
such, standard frequency counting techniques,
although adequate to measure beat frequencies
between state-of-the-art optical clocks, cannot
yield the required precision for our micro-
waves. Achieving this level of phase resolution,
and maintaining it overseveral hours, was ac-
complished in the following ways. First, we
utilized microwave amplifiers with low flicker
noise, and we routed signals with temperature-
insensitive cabling. This gave us an output
withamplepower(~10mW)withoutsacrific-
ing stability. Second, by frequency-mixing the
10-GHz outputs, we shifted the measurement
to a 1.5-MHz carrier. This reduced the require-
ments on the fractional stability that we had to
measureby~7000(10GHz/1.5MHz).Bydigi-
tally sampling the 1.5-MHz carrier, the phase
difference between the 10-GHz outputs could
betrackedwithhighresolution.Anotherad-vantage of this measurement scheme is that the
high phase resolution is achievable without
requiring the two sources to oscillate at exactly
the same frequency. This gives our measure-
ment system some dexterity in comparing high
stability signals from independent sources.
Optical and microwave phase fluctuations,
continuously recorded over 44,000 s, are shown
in Fig. 2. In Fig. 2A, the phase fluctuations of
the optical clock have been scaled by a factor
equal to the optical-to-microwave frequency
ratio (nearly 26,000) to illustrate the extremely
high fidelity in the optical-to-microwave trans-
fer.Therelativephasecanalsobeexpressedas
a timing fluctuation and is bounded by ±30 fs
for both optical and microwave signals. Also
showninFig.2Aisthepoint-by-pointdifference
between optical and microwave measurements,
limited to root mean square (RMS) fluctuations
of 60 mrad, corresponding to a RMS relative
timing fluctuation of only 900 as. This implies
that optical clocks with even higher stability
can be converted to microwave signals with-
out loss of fidelity. The strong correlation be-
tween the optical and microwave phases is
shown in Fig. 2B. The degree of correlation is
quantified by the correlation coefficient, rang-
ing from zero for completely uncorrelated phases
to a maximum value of 1 for complete linear cor-
relation between optical and microwave phase
( 27 ). The calculated correlation coefficient for
the data in Fig. 2 is 0.998. Such femtosecond-
level, high coherence optical-to-microwave con-
version opens up the possibility of connecting
distant optical clocks with a microwave link.
Currently, these clocks can be linked optically
through fiber or over free space ( 28 ), enabling
state-of-the-art clock comparisons and syn-
chronization. Free-space optical links are par-
ticularlyusefulforthemanysituationswhere
a dedicated fiber link is not available but can
become ineffective because of poor weather or
dusty conditions. The lower loss of microwave
transmission could prove advantageous under
such conditions by providing a link that would
be impossible to maintain optically.
Whereas fluctuations in the phase provide
all the frequency and timing stability informa-
tion of an oscillator, the fractional frequency
instability is the more typical performance
benchmark. Figure 3 displays the fractional
frequency instabilities derived from the same
44,000-s duration phase measurements shown
in Fig. 2A. The frequency stability of the derived
microwaves followed that of the optical clocks
precisely, ultimately yielding an absolute frac-
tional frequency instability of 1 × 10−^18 .Thisis
100 times more stable than the Cs fountain
clocks that currently serve as the best realiza-
tion of the SI second. The short-term stability
also exceeds that of other microwave sources,
the best of which are microwave oscillators
based on whispering gallery mode resonances
in cryogenically cooled sapphire ( 29 ). ThereNakamuraet al.,Science 368 , 889–892 (2020) 22 May 2020 3of4
Fig. 3. Fractional frequency stability comparison of state-of-the-art sources.The microwave and optical
signals locked to the Yb clocks are reported in terms of the total Allan deviation, with error bars representing
1 sconfidence intervals. The white frequency noise asymptote is 1: 6 10 ^16 =
ffiffi
tp. Additionally, the residual
instability of the optical-to-microwave link is well below state-of-the-art optical clocks ( 7 , 29 – 31 ). For all plots,
the frequency stability is given by the Allan deviation.
RESEARCH | REPORT